For \(i,j=1,\dots ,N\) , the integrand on the right-hand side of (REF ) has singularities coming from the factor \(\Gamma (\xi _j-Iw_i)\)  at each point of the form \(\xi _j=Iw_i-\nu \)  for \(\nu \in \mathbb {Z}_{\ge 0}\)  (where \(\mathbb {Z}_{\ge 0}\)  stands for the set of non-negative integers). By shifting each contour to the left to \(-\infty \)  we will pick up each of these singularities, and the result is that the integral will equal the sum of the associated residues (The fact that the contours of integration can be deformed in this way follows from our assumption on \(f\)  and the known asymptotics of the Gamma function, which give \(c_1 e^{-\frac{\pi }{2}|\mathrm {Im}(z)|}|\mathrm {Im}(z)|^\eta \le |\Gamma (z)|\le c_2 e^{-\frac{\pi }{2}|\mathrm {Im}(z)|}|\mathrm {Im}(z)|^\eta \)  for \(\mathrm {Re}(z)\)  in some finite interval and some \(c_1,c_2>0\)  and \(\eta \in \mathbb {R}\) , see, e.g., [1]};
observe also that, thanks to the factor \(u^{\sum _i(Iw_i-\xi _i)}\) , there is no pole at \(-\infty \) ).
Each of these residues is evaluated at a point of the form \((\xi _1,\dots ,\xi _N)=(Iw_{m_1}-\nu _{1},\dots ,Iw_{m_N}-\nu _{N})\)  for some \((m_1,\dots ,m_N)\in \lbrace 1,\dots ,N\rbrace ^N\)  and \(\nu _{i}\ge 0\)  for \(i=1,\dots ,N\) , so the right-hand side of (REF ) equals
\(\frac{1}{N!}\sum _{(m_1,\dots ,m_N)\in \lbrace 1,\dots ,N\rbrace ^N}\sum _{\nu \in (\mathbb {Z}_{\ge 0})^N}u^{\sum _i(Iw_i-Iw_{m_i}+\nu _i)}\frac{(-1)^{\sum _i\nu _i}}{\prod _i\nu _i!}f(w_{m_1}+I\nu _1,\dots ,w_{m_N}+I\nu _N)\\\qquad {}\times \prod _{i\ne j}\Gamma (Iw_{m_j}-\nu _j-Iw_i)\prod _{i\ne j}\frac{1}{\Gamma (Iw_{m_i}-\nu _i-Iw_{m_j}+\nu _j)}.\) 
